Performance analysis and design of circular high‐strength concrete‐filled double‐skin aluminum tubular short columns under axial loading

This study investigates the performance and design of high‐strength circular concrete‐filled double‐skin aluminum tubular (CFDAT) columns under axial loading. A new fiber element (FBE) model incorporating a new lateral confinement model is developed that considers the confinement effects and material nonlinearities. A new strength degradation factor is proposed for determining the postpeak behavior of the confined concrete in CFDAT circular columns. Existing experimental results are used to validate the accuracy of the predicted ultimate strength and axial load‐strain ( P−ε ) curves of CFDAT columns under axial loading. A comparison of the predictions of the ultimate strength and P−ε curves of CFDAT columns using the proposed lateral confinement model is made against the prediction using the three‐dimensional finite element modeling and the existing lateral pressure model of CFDAT columns proposed by other researchers. The performance of the CFDAT columns under axial loading is investigated using a detailed parametric study. The accuracy of existing empirical formulas given in various design standards for conventional concrete‐filled steel tubular columns as well as given by another researcher in predicting the ultimate strength of CFDAT columns is examined. Finally, a simple design formula is proposed and validated against the experimental and numerical results obtained from this study. It is found that the proposed FBE model and the simplified model developed in this study can accurately predict the performance of CFDAT columns under axial loading.

6][7][8][9] The flexural stiffness and fire resistance of CFDST columns are higher compared to CFST columns. 5n addition, due to the central void, CFDST columns are lightweight compared to that CFST columns and offer dry space for engineering installation.In terms of crosssection, a circular CFDST column provides more uniform confinement to the confined concrete thus offering higher strength and ductility compared to a rectangular or square CFDST column. 10The circular CFDST columns were reported to be used in transmission towers and have potential applications as bridge piers and to support other marine structures. 5However, constantly being exposed to the marine environment will cause the steel tubes to corrode which may significantly affect the structural performance of CFDST columns.
In protecting from being corroded, corrosion-resistant materials such as aluminum can be used as the outer tube in CFDST columns and such a column can be referred to as a concrete-filled double-skin aluminum tubular (CFDAT) column.In comparison to the same material strength, aluminum is about 35% less in weight compared to carbon steel and offers significant corrosion resistance compared to carbon steel. 11,12Thus, the weight of the CFDST columns can be reduced by utilizing aluminum as the outer tube (CFDAT columns).However, due to the lower Young's modulus of aluminum which is about one-third of Young's modulus of carbon steel, for an identical cross-section with the same material properties of steel and concrete, the ultimate strength of a CFDAT column will be lower than a CFDST column.High-strength concrete (HSC) can be used to increase the ultimate strength of CFDAT columns but this may reduce the ductility of the columns.However, until today, limited investigations have been carried out on the performance analysis and design of high-strength CFDAT columns under axial loading, 11,13,14 which results in a limited understanding of their structural performance and development of design guidelines for structural engineers by existing design codes such as Eurocode 4, 15 AS 5100.6.6, 16 AISC 360-16-16 17 and ACI 318-19. 180][21] However, tests on circular CFDAT columns were only reported by Zhou and Young. 13Zhou and Young 13 conducted a series of tests to investigate the influences of the key column parameters on the behavior of circular CFDAT short columns, where the concrete strength was varied up to 106 MPa.Test results showed that the initial stiffness and ultimate strengths of the CFDAT columns were enhanced with the increase in concrete strength; however, columns with higher concrete strength exhibited brittle failure compared to that of normal strength concrete.
3][24][25][26][27][28][29][30] The only numerical models for circular CFDAT columns were developed by Zhou and Young, 13 and Patel et al. 11 Zhou and Young 13 developed finite element (FE) models using ABAQUS where the composite action between the concrete and aluminum was captured using the Drucker-Prager model defined in the ABAQUS library.Patel et al. 11 proposed a new lateral pressure formula for the confined concrete model similar to the one used by Liang, 31 and Hu and Su 32 for analyzing CFDST columns composed of carbon steel which was only the function of the ratios of tube diameter to the wall thickness of external and internal steel tubes (D o /t o and D i /t i ).As discussed by Ahmed et al. 5 and Yan and Zhao, 29 the absence of the important column parameters on the existing concrete confinement models of CFDAT columns such as the hollow ratio or the confinement factor may result in an inaccurate estimation of the lateral stress of such a column.Furthermore, the postpeak behavior of the confined concrete of CFDAT columns was determined using the strength degradation parameter proposed by Liang 31 based on the work of Hu and Su, 32 originally proposed for CFDST columns composed of carbon steel.In addition, the formula was developed based on the limited test data, 33,34 where the maximum concrete strength was limited to 63.4 MPa, and the formula proposed was only the function of the ratios of tube diameter to the wall thickness of external and internal steel tubes (D o /t o and D i /t i ).6][37] The absence of the parameter of concrete strength in the existing formula used to calculate the postpeak behavior of confined concrete of CFDAT columns by Patel et al. 11 may not provide an accurate response for such a column.Therefore, a more accurate confinement model is needed to be proposed to accurately predict the behavior of circular CFDAT columns.9][40] However, Imran and Pantazopoulou 41 and Lan and Guo 42 stated that confined concrete strength was essentially independent of the shape of the loading path.4][45][46][47][48][49] Lai et al. 45 proposed cementitious fillers and inert fillers to improve the flowability and passing ability of concrete in composite columns.Wong and Kwan 48 proposed a wet packing method that can be used to measure the packing density and water demand instead of indirect methods.
This study develops a fiber element (FBE) model for the performance analysis of axially loaded circular highstrength CFDAT short columns.A new formula to calculate the lateral stress to the confined concrete and a novel strength degradation coefficient for determining the postpeak characteristics of confined concrete in the studied CFDAT columns are proposed.In addition, the applicability of current design standards of CFST columns in designing CFDAT columns is evaluated.A novel simple design formula is developed and validated against the test and numerical results to predict the compressive strength of the CFDAT columns under axial loading.

| General
1][52] In the FBE model, the cross-section of the CFDAT column is divided into fine fiber elements, as illustrated in Figure 1.Each fiber element is assigned with the material properties of aluminum or concrete using the material uniaxial stress-strain relationships of concrete and aluminum presented in Sections 2.2 and 2.3, respectively.The fiber stresses of the cross-section of CFDAT columns are determined from corresponding fiber strains.The axial load of the column is determined using the stress resultant using Equation (1). Figure 2 illustrates the computer flowchart for calculating P À ε curves of CFDAT columns.
where N refers to the axial load of the columns; σ ao,i and σ ai,k stand for the stresses of fiber element of the outer and inner tube, respectively; σ c,j symbolizes the stress of fiber element of sandwiched concrete; A ao,i and A ai,k represent the area of fiber element of the outer and inner tube, respectively; A c,j stands the area of fiber element of sandwiched concrete; n ao , n sc and n ai is the total number of fiber elements of outer and inner tubes and infilled concrete, respectively.[55][56] F I G U R E 1 Typical fiber mesh of the cross-section of a circular CFDAT column.
The ductility index value (DI) of an axially loaded CFDAT column is calculated as [52][53][54][55][56][57] : where ε u refers to the strain in the postyield stage when the axial load fell to 90% of the ultimate compressive load, or the ultimate strain when the column appears a strengthened postyield behavior.The yield strain ε y was calculated as ε 0:75 =0:75, where ε 0:75 denotes the strain when the axial load reaches 75% of the ultimate compressive load. 52-57

| Material laws of confined concrete
The idealized stress-strain curves of concrete employed in this study are shown in Figure 3, which comprises two curves: the rising segment OA (0 ≤ ε c ≤ ε 0 cc ) and declining segment AB (ε c > ε 0 cc ).The formula developed by Mander et al. 58 is used to calculate the stress of concrete (σ c ) in the rising segment of stress-strain curves written as: in which ε c denotes the strain of concrete at σ c ; f 0 cc is the compressive strength of confined concrete and ε 0 cc is the corresponding strain; E c stands for the elastic modulus of the infilled concrete, given by ACI 318-19 18 as: where γ c is the reduction factor of the compressive strength of unconfined concrete to consider the effects of the column size, load rates, concrete quality and proposed by Liang 31 : where t c is the wall thickness of concrete filled between external and internal aluminum tubes, calculated as where D o and D i are the diameters of external and internal aluminum tubes, respectively, and t o is the wall thickness of the external aluminum tube.The formulas proposed by Mander et al. 58 and Richart et al. 59 are given in Equations ( 7) and ( 8) with the factor γ c given by Liang 31 are utilized to calculate f 0 cc and ε 0 cc .
where ε 0 c is the strain at f 0 c , calculated using the formula proposed by De Nicolo et al. 60 : In Equations ( 7) and ( 8), f rp is the lateral stress to the confined concrete of CFDAT columns inserted by aluminum tubes.The accurate estimation of the lateral confining pressure is important in simulating the performance of CFDAT columns under axial compression.A new lateral stress model is proposed based on the test results of Zhou and Young. 13In doing so, first, the section capacities of the aluminum tubes were subtracted from the ultimate compressive strength of the tested columns.The lateral confining stress of each of the tested columns was then calculated from Equation (7).A formula to calculate f rp is proposed based on the linear regression analysis as: in which D o =t o is the diameter to wall thickness ratio of outer aluminum tube, χ stands for the hollow ratio calculated as χ ¼ D i =D o À 2t o ; η represents the confinement coefficient reflecting concrete confinement effect, proposed by Yan et al. 8 as: F I G U R E 3 Idealized stress-strain relationships of confined and unconfined concrete.
As discussed earlier, the absence of the parameter of concrete strength in the existing formula used to calculate the postpeak behavior of confined concrete of CFDAT columns may not provide an accurate response for such a column.Patel et al. 11 used the formula proposed by Liang and Fragomeni 61 in calculating the postpeak curves of concrete in CFDAT columns as illustrated in Figure 3, calculated as: where ε cu represents the ultimate strain and β c is the strength degradation parameter calculated as (β c ¼ f cr =f 0 cc ); f cr is the residual strength of concrete.3][64][65][66] The β c is calculated as: in which the k 3 is a factor proposed by Hu and Su 32 as: According to the aforementioned formulas, β c only depends on D o /t o and D i /t i ratios.However, test results show β c is affected by the f 0 c , D o =t o and D i =t i ratios.In the FBE model developed, the declining segment AB (ε c > ε 0 cc ) of the stress-strain curves of concrete is calculated using the following formula developed in this study: where ε ct is the strain at the inflection point, calculated using the formula given by Lim and Ozbakkaloglu 37 with a slight modification incorporating the strength degradation coefficient β c and parameter γ c given in Equation ( 6): The strength degradation coefficient β c was estimated using the trial and error methods to best fit the axial load-strain curves of each of the tested columns reported by Zhou and Young, 13 Wang et al. 20 and Yan et al. 7 A formula based on the regression analysis is developed as: The R 2 is calculated as 0.948 as illustrated in Figure 4, which confirms a good correlation between the predictions and test results.

| Material laws of aluminum
The formulas given by Ramberg and Osgood, 67 given in Equation ( 18) are commonly used to calculate the stressstrain relationships of aluminum.
in which E o is Young's modulus of aluminum; σ 0:2 is 0.2% proof stress; n is the strain hardening factor (n ¼ ln 20 ð Þ= ln σ 0:2 =σ 0:01 ð Þ ); σ 0:01 is the 0.01% proof stress at the strain ε p ¼ 0:01%.However, Equation ( 18) overestimates the true stress values for the stress level greater than σ 0:2 . 68Recently, Gardner and Nethercot 68 proposed a two-stage Ramberg- Osgood model, which is accurate until the strain reaches 5%.Mirambell and Real, 69 and Rasmussen 70 proposed a modified Ramberg-Osgood stress-strain relationship, and its application scope was stretched into the ultimate tensile strength σ u .In general, the calculations of the models given above are based on stress to compute strain.Abdella 71 proposed an inverted version of the stressstrain relationship where the stress is calculated through a given strain.Considering the outer and inner tubes of the CFDAT columns are under biaxial stresses owing to the confinement effect, the 0.2% proof stress of the tubes is reduced by 10% based on the existing studies. 5,38The stress-strain relations proposed by Abdella 71 illustrated in Figure 5 that consider the strain hardening effects can be calculated using the following equation: where σ a and ε a are the aluminum stress and its corresponding strain; σ au is the ultimate tensile stress of the aluminum; ε au is the ultimate tensile strain at the stress σ au ; ε 0:2 stands for the strain at the proof stress σ 0:2 ; ε 0 0:2 represents the strain when the stress is taken as 0.9 σ 0:2 in ascending stage.The expression of the strain ε 0 0:2 is given by: In Equation ( 20), r, r 2 , r Ã , p, and p Ã are the material constants, calculated as: in which E 0:2 is the tangent stiffness at the stress 0:9σ 0:2 , calculated as: The expression of the slope E u in Equation ( 26) is defined as: where m is the poststrain hardening exponent, and calculated as m ¼ 1 þ 3:5 0:9σ 0:2 =σ au ð Þ .

| EVALUATION OF THE ACCURACY OF THE FIBER ELEMENT MODEL
The proposed FBE model incorporating the accurate material models is validated by comparing the predicted ultimate strength and P À ε curves against the test results reported by Zhou and Young, 13 as well as with predictions of three-dimensional finite element (3D FE) modeling developed by Zhou and Young, 13 and the lateral confinement model for CFDAT columns proposed by Patel et al. 11 The details related to the development of the 3D FE model are given by Zhou and Young. 13A summary of the test data of CFDAT columns reported by Zhou and Young 13 is given in Table 1, where σ 0:2o and The stress-strain relationships of aluminum.
σ 0:2i are the 0.2% proof stresses for outer and inner tubes, respectively and χ is the hollow ratio.

| The P À ε curves
The numerical results are further validated by comparing the predicted P À ε curves with the experimental results as well as with those obtained from the 3D FE model and the lateral confinement model proposed by Patel et al. 11 It can be seen in Figure 6 that, in the elastic stage, the predictions of the proposed FBE model, 3D FE model, and the lateral confinement model proposed by Patel et al. 11 are well in agreement with the P À ε curves of the tested columns except for specimens C4C2-C70 and C5C1-C100 with a small discrepancy.This discrepancy is mainly attributed to the uncertainty of existing infilled concrete used in numerical analysis.In the postpeak stage, the predicted curves of the proposed FBE model correlate well with the P À ε curves of specimens.

| PERFORMANCE OF CIRCULAR CFDAT SHORT COLUMNS UNDER AXIAL LOADING
The proposed FBE model was employed to examine the influences of the diameter ratio (D i /D o ), the ratio of the diameter-to-tube thickness (D/t) of outer and inner tubes, proof stress (σ 0.2 ) of external and internal aluminum tubes, and the concrete strength (f 0 c ) on the compressive behavior of the CFDAT columns.In this parametric study, six series of full-size CFDAT columns are T A B L E 3 Comparisons of the calculated lateral stress on the core concrete confined by circular carbon steel and aluminum tubes using various lateral pressure models.

Specimen
N u,exp (kN) Liang 31 Patel et al. 11 Proposed method, Equation ( 10)  4. The elastic modulus of the internal and external aluminum tubes is assumed to be 69 GPa, and the ultimate axial strain of concrete ε cu is specified as 0.04.The non-linearity index n for aluminum is taken as 5.0.

| Influences of the diameter ratio (D i /D o )
The effects of the D i /D o ratios on the initial stiffness, ultimate axial strength and ductility of the studied CFDAT columns were evaluated by using the FBE model developed.The details of the D i /D o ratios are shown in Series 1 in Table 4.As listed in Series 1, the D i /D o ratios were defined by changing the outside diameter of the outer aluminum tube whereas keeping its wall thickness constant.The P À ε curves and ductility index for the CFDAT columns are depicted in Figure 7a,b, respectively.From Figure 7a

| Influences of the tube diameter to wall thickness ratio (D i /t i ) of inner aluminum tube
In this section, using the proposed FBE model, two cases were employed to ascertain the effects of the D i /t i ratios on the initial stiffness, ultimate axial strength, and ductility of the studied CFDAT columns.For the first case, the wall thickness was kept constant by changing the diameter of the internal aluminum tube, as shown in Series 3 in Table 4.The P À ε curves and ductility index for the studied CFDAT columns with different D i /t i ratios by varying the tube diameter are depicted in Figure 9a,b, respectively.From Figure 9a, it can be found that increasing the D i /t i ratios decrease the initial stiffness of the studied CFDAT columns.With the increase of the D i /t i ratios from 15 to 20, 25, and 30, the compressive bearing capacities of the studied CFDAT columns reduce by 3.8%, 9.2%, and 18.1%, respectively.The reason for this is primarily that the area of infilled concrete is decreased while the area of the inner aluminum tube increased.Moreover, from Figure 9b, it can be noticed that increasing the D i /t i ratios increases the ductility indices of the studied CFDAT columns.The ductility indices for the D i /t i ratios of 15, 20, 25, and 30 are 15.46, 15.71, 16.76,  and 17.14, respectively.This is due to the increase in the steel area.
As shown in Series 4 in Table 4, the second case was to change the wall thickness of the inner aluminum tube while maintaining its diameter unchanged.For Series 4, the P À ε curves and ductility index for the studied CFDAT columns for varying D i /t i ratios by varying the tube thickness are illustrated in Figure 9c,d, respectively.From Figure 9c, it can be found that increasing the D i /t i ratios by changing the wall thickness of the internal aluminum tube generates a negligible effect on the initial stiffness of the studied CFDAT columns.The numerical results indicate that with the increase of the D i /t i ratios from 15 to 20, 25, and 30, the compressive bearing capacities of the studied CFDAT columns decrease by 1.9%, 5.2%, and 6.4%, respectively.Moreover, as shown in Figure 9d, the ductility indices of the studied CFDAT columns reduce as the D i /t i ratios increase.For the D i /t i ratios of 15, 20, 25, and 30, the ductility indices are 2.58, 2.37, 2.29, and 2.25, respectively.This is primarily due to the reduction of the steel area.

| Influences of the concrete strength (f 0
c ) The effects of f 0 c on the initial stiffness, ultimate axial strength, and ductility of the studied CFDAT columns were investigated.In Series 5, the compressive strengths of infilled concrete were designed as 55, 75, 95, and 115 MPa.The P À ε curves and ductility index of the studied CFDAT columns with different concrete strengths are depicted in Figure 10a,b, respectively.From Figure 10a, it can be noticed that the initial stiffness of the studied CFDAT columns is considerably increased as the concrete strength increases.With the increase of the concrete strengths from 55 to 75, 95, and 115 MPa, the compressive bearing capacities of the studied CFDAT columns increased by 26.5%, 53.3%, and 80.2%, respectively.The utilization of HSC greatly enhances the cross-sectional load resistance of CFDAT columns.This result suggests that it is an effective method to improve the compressive bearing capacities of the studied CFDAT columns by infilling HSC.From Figure 10b, however, it can be observed that the ductility indices of the studied CFDAT columns are reduced as the concrete strength increases.The ductility indices for CFDAT columns with the concrete strengths of 55, 75, 95, and 115 MPa are 4.60, 2.35, 1.93, and 1.62, respectively.This is due to the brittle nature of HSC.

| Influences of the proof stresses of external and internal aluminum tubes
The CFDAT columns listed in Series 6 in Table 4 were utilized to investigate the effects of the proof stress of aluminum tubes on their P À ε curves and ductility index.In Series 6, the proof stresses of external and internal aluminum tubes were specified as 110, 160, and 240 MPa.The P À ε curves and ductility index of the studied CFDAT columns with different aluminum proof stresses are illustrated in Figure 11a,b, respectively.From Figure 11a, it can be found that the initial stiffness of the studied CFDAT columns is insignificantly affected by the aluminum proof stress.With the increase of the aluminum proof stresses from 110 to 160 and 240 MPa, the crosssectional load resistance of the studied CFDAT columns increased by 8.3% and 18.8%, respectively.On the other hand, from Figure 11b, the ductility indices of the studied CFDAT columns increase as the aluminum proof stress increases.The ductility indices for CFDAT columns with the aluminum proof stresses of 110, 160, and 240 MPa are 2.51, 2.55, and 2.93, respectively.

| EVALUATION OF THE ACCURACY OF THE VARIOUS DESIGN CODES
Existing design standards have not provided design guidelines to obtain the compressive strengths of the studied CFDAT short columns.This section investigates the applicability of the existing design guidelines for CFST columns specified in Eurocode 4 (EC4), AS 5100.6,AISC 360-16-16, and ACI 318-19 in designing CFDAT columns.The existing design guidelines for CFST columns modified for predicting the bearing capacities of CFDAT short columns are presented in Table 5.The original limit D=t ≤ 90 235=f y on the slenderness of the outer steel tube specified in EC4 was modified for aluminum tubes as D=t ≤ 90 235=σ 0:2 ð ÞE o =210, 000 ð Þconsidering the differences in material properties.The yield stress f y in the original limit on the slenderness of the outer steel tube was replaced by the 0.2% proof stress of aluminum.The effective cross-sectional area for aluminum tubes was calculated in computing the cross-sectional load resistance of specimens when the normalized slenderness of specimens was larger than the normalized slenderness limit λ L specified by EC4, AS 5100.6 and ACI 318-19.For EC4, an effective area formula for steel tube developed by Chan and Gardner 72 based on the one in BS5950-1 73 was modified considering the differences in material properties.The effective area of AS 5100.6 and ACI 318-19 were determined by the equations suggested by AS/NZS 4673 74 and SEI/ASCE-8-02. 75Unlike EC4, AS 5100.6 and ACI 318-19, AISC 360-16 determines the cross-sectional load resistance of specimens by dividing the composite sections into a compact, noncompact and slender sections instead of using the effective area.
The comparisons of the predicted ultimate strength of CFDAT columns using various design models against the test ultimate strengths of the columns tested by Zhou and Young 13 as well as the ultimate strength of the columns utilized for parametric study are shown in Tables 6 and  7, respectively.It is seen that the predictions of EC4, AISC 360-16 and AS 5100.6 deviated greatly from the experimental results with an unsafe trend.However, ACI 318-19 yields significantly conservative predictions for the ultimate strengths of the columns.This is mainly because ACI 318-19 does not account for the effects of concrete confinement.In general, the current design standards for CFST columns cannot accurately predict the ultimate strengths of CFDAT columns.This is due to the differences in material properties of carbon steel and aluminum.The existing formulas specified in existing design codes are developed based on extensive test data on CFST columns.Therefore, a new empirical model should be developed based on the test data of CFDAT columns to accurately predict their ultimate strength.

| SIMPLIFIED DESIGN MODEL
Similar to previous investigations, [76][77][78][79] based on the lateral confinement model developed in this study as well as using the numerical results obtained using the FBE model developed, a novel simplified design formula is proposed to predict the compressive strengths of the axially loaded circular CFDAT short columns written as: where A ao and A ai are the cross-sectional area of outer and inner aluminum tubes, respectively; A c is the crosssectional area of the infilled concrete.Parameters γ c and f rp,pro are determined using Equations ( 6) and (10), respectively.γ ao and γ ai are the strength factors of the outer and inner aluminum tubes considering the strain hardening, hoop tensile stresses, and imperfections of the aluminum tub.Based on the test and numerical results, the expressions for γ ao and γ ai are proposed in this study as: Patel et al. 11 validated the design equation similar to Equation (29) proposed by Liang 31 for CFDST columns to calculate the ultimate strength of CFDAT columns by replacing the lateral stress proposed for CFDAT columns, given in Equation (32).MPa) F I G U R E 1 1 Influences of the aluminum proof stresses on the p À ε curves and ductility of CFDAT columns.
T A B L E 5 Formulas to calculate the ultimate strength of circular CFDAT short columns using various design codes.

Design codes
- f rp,Patel ¼ 10:0445 À 0:3090 The validation of the proposed formula given in Equation ( 29) is performed by comparing the predicted ultimate strength with the test ultimate strengths of the columns tested by Zhou and Young 13 as well as the ultimate strength of the columns utilized for the parametric study given in Tables 6 and 7, respectively.The ultimate strength using the formula used by Patel et al. 11 is also compared with the test and numerical strength of CFDAT columns to verify the accuracy of the two design models.It can be seen the mean of the prediction-to-test ultimate strength using the design model proposed is equal to the unity.The proposed model can accurately predict the both test and numerical ultimate strength of the columns.However, Patel et al. 11 significantly overestimate the compressive strengths of CFDAT columns investigated numerically in this study, as can be seen in Table 7.The mean of the prediction-to-test ultimate strength using Patel et al. 11 is calculated as 1.062.The   11 The influences of key parameters on the performance of CFDAT columns are examined using a detailed parametric study.The accuracy of various empirical formulas for designing CFDAT columns including the one proposed in this study is examined.It is found that the proposed FBE model can predict the ultimate compressive strength and the P À ε curves of

F I G U R E 4
Validation of the formula proposed for estimating β c (Equation17).

7
Influences of the D i /D o ratios on the p À ε curves and ductility of CFDAT columns.increase.For the D o /t o ratios of 30, 40, 50, and 60, the ductility indices are 4.10, 3.54, 2.98, and 2.66, respectively.

8 9
Influences of the D o /t o ratios on the p À ε curves and ductility of CFDAT columns.Influences of the D i /t i ratios on the p À ε curves and ductility of CFDAT columns.

1 0
Influences of the concrete strengths on the p À ε curves and ductility of CFDAT columns.

T A B L E 6 13 Specimens
Evaluation of the accuracy of the various design formulas in predicting the ultimate strength of circular CFDAT short columns tested by Zhou and Young.

T A B L E 7
Evaluation of the accuracy of the various design formulas in predicting the ultimate strength of circular CFDAT short columns used for the parametric study.

Table 2
presents the comparisons of the test ultimate strength of CFDAT columns with the predictions by the proposed FBE model, 3D FE model and the lateral confinement model proposed by Patel et al. 11 where it can be observed that the numerical strengths of the 3D FE model and the lateral confinement model proposed by Patel et al.
11are, on average, 2.1% and 2.0% higher than the test strengths, respectively.The mean of the predictions of the proposed FBE model to test strengths is 99.7%, which is closer to unity.This indicates that the 3D FE model and the lateral confinement model proposed by Patel et al.11generate unconservative

Table 3
13mmary of the test results of circular CFDAT short columns tested by Zhou and Young.13 11aluates the accuracy of the existing lateral pressure models proposed by Liang31originally proposed for CFDST columns with carbon tube and the T A B L E 1 eral pressure model for carbon steel and the lateral pressure model developed for CFDAT columns by Patel et al.11are calculated as 1.016 and 1.020, respectively.On the contrary, the lateral pressure model proposed in this study obtain a mean prediction-to-test ultimate strength of 0.997.
Comparisons of predicted ultimate strength using the proposed FBE method, FE and Patel's models against the test ultimate strength of CFDAT columns.
T A B L E 2 and C5C1-C100 otherwise shown in the test.This is because the strength degradation formula used byPatel  et al. 11in predicting the residual strength of concrete only considered the influences of the D o /t o and D i /t i ratios and does not consider the effects of concrete strength.On the other hand, the predicted P À ε curves using the proposed FBE model show good accuracy than the existing models.
Comparisons of the predicted p À ε curves of CFDAT columns against the test results.columns in Series 2 utilizing the proposed FBE model.For Series 2, the D o /t o ratios were changed by varying the thickness of the outer aluminum tube but keeping its diameter constant.The P À ε curves and ductility index for the studied CFDAT columns with different D o /t o ratios are depicted in Figure8a,b, respectively.From Figure8a, it can be observed that the D o /t o ratio has an insignificant influence on the initial stiffness of the studied CFDAT columns.With increasing the D o /t o ratios from 30 to 40, 50, and 60, the compressive bearing capacities of the studied CFDAT columns decrease by 11.0%, 16.8%, and 18.3%, respectively.Additionally, from Figure8b, it can be seen that the ductility indices of the studied CFDAT columns decrease as the D o /t o ratios T A B L E 4 Summary of the geometry and material properties of axially loaded circular CFDAT short columns used in the parametric study.
, it can be noticed that the initial stiffness of the CFDAT columns increases with the decrease of the D i /D o ratio.When the D i /D o ratios decrease from 0.4.2 | Influences of the tube diameter to wall thickness ratio (D o /t o ) of outer aluminum tubeThe effects of the D o /t o ratios on the behavior of the studied CFDAT columns were also studied by analyzing incorporating accurate material models of aluminum and confined concrete.A novel lateral pressure formula, as well as a new strength degradation coefficient for determining the postpeak characteristics of confined concrete in the studied CFDAT columns are proposed based on the test results.The accuracy of the FBE model developed is validated against the test results as well as against the predictions by 3D FE modeling and the lateral confinement model proposed byPatel et al.
7 | CONCLUSIONSThis paper develops a FBE model for the nonlinear analysis of circular high-strength CFDAT short columns under axial compression CFDAT columns accurately compared to others, particularly for columns made of HSC.A mean ratio of the prediction-to-test ultimate strength is obtained as 1.00.The parametric study shows that increasing the hollow ratio by reducing the outer tube diameter remarkably reduces the initial stiffness and cross-sectional load resistance, whereas increases the ductility of CFDAT short columns.While aluminum proof stress has a moderate influence on the ultimate strength of CFDAT columns, the strength and ductility of CFDAT columns are significantly influenced by concrete strength.In terms of design, existing empirical formulas cannot accurately predict the ultimate strength of CFDAT columns.However, the simplified model developed in this study can provide a more accurate design estimation of such a column.The limitation of the numerical models are as follows: the